Optimal strategy of coupon subset collection when each package contains half of the coupons

The coupon subset collection is a generalization of the classical coupon collection, where instead of selecting (with replacement) a single coupon, a subset of at most k coupons (known as a “package”) is selected in each round. In this paper, we study how to design the pool of packages and assign probabilities to the packages, so as to minimize the expected number of rounds to collect all n distinct coupons. When k divides n, a seemingly optimal strategy is to choose a pool of non-intersecting packages, and assign equal probability to each package in the pool. We prove the optimality of this strategy when the size of the package is half the number coupons, that is, n=2k. •We consider the well-known problem of coupon subset collection.•Optimal solutions are known for only two special cases: (n,k)=(4,2) and (n,k)=(6,2).•We prove that a simple strategy is optimal when k=n/2 for any k.•We prove that it is the unique optimal up to permutation of the coupons.